Question

Find the point of intersection between the line and the plane. line: \(\langle 4,1,0\rangle+t\langle 1,0,-1\rangle\) plane: \(3 x+y-2 z=8\)

Short answer

The point of intersection is (3, 1, 1).

Step-by-step solution

Express the Line in Parametric Form

The line is given in vector form as \( \langle 4,1,0\rangle + t\langle 1,0,-1\rangle \). Let's convert this into parametric equations for \(x\), \(y\), and \(z\):- \( x = 4 + t \)- \( y = 1 \)- \( z = 0 - t \)

Substitute into Plane Equation

Substitute the parametric equations of the line into the plane equation \(3x + y - 2z = 8\). Replace \(x\), \(y\), and \(z\) with their parametric expressions:\[3(4 + t) + (1) - 2(0 - t) = 8\]

Simplify and Solve for t

Simplify the expression from the substitution:\[3 \times 4 + 3t + 1 + 2t = 8\]This simplifies to:\[12 + 3t + 1 + 2t = 8\]Combine like terms:\[13 + 5t = 8\]Solve for \(t\) by subtracting 13 from both sides:\[5t = 8 - 13\]\[5t = -5\]Then divide by 5:\[t = -1\]

Find the Intersection Point

With \(t = -1\), use the parametric equations to find the point of intersection:- For \(x\): \( x = 4 + (-1) = 3 \)- For \(y\): \( y = 1 \)- For \(z\): \( z = 0 - (-1) = 1 \)Thus, the point of intersection is \((3, 1, 1)\).

Key concepts

Line and Plane Intersection

Understanding the point of intersection of a line and a plane is an essential concept, especially in geometry and vector calculus. A line in space can intersect a plane at a single point, or it can be entirely on the plane, or not intersect at all if they are parallel and do not coincide.
To find where, or if, a line intersects a plane, we substitute the line's parametric equations into the plane's equation. This substitution gives us a way to analyze the interaction between these geometric forms, leading to the solution for the specific point of intersection. In the given exercise, we determine the point where the line represented by its vector form meets the defined plane intersection.

Parametric Equations

Parametric equations are a way to express a line in space using a parameter—often denoted as \(t\)—that varies over the set of real numbers. In the context of a line, this means:

• Each point on the line is a result of plugging a specific \(t\) value into the parametric equations.
• For the given line \(\langle 4,1,0\rangle+t\langle 1,0,-1\rangle\), the parametric equations are:
• \(x = 4 + t\)
• \(y = 1\)
• \(z = 0 - t\)

These equations tell us precisely how \(x\), \(y\), and \(z\) change as \(t\) changes, offering a dynamic view of how the line is positioned and traverses space.

Substitution Method

The substitution method is a crucial step used when determining intersections in geometry. It involves replacing variables with known expressions or values to simplify equations.
In the given exercise, we substitute the parametric expressions for \(x\), \(y\), and \(z\) from the line into the plane's equation \(3x + y - 2z = 8\). This substitution focuses our problem from three-dimensional space to a single variable, \(t\), that we can solve more straightforwardly:
\[3(4 + t) + 1 - 2(0 - t) = 8\]
This substitution method is the gateway for simplifying the interaction between the line and the plane, allowing us to zero in on the exact conditions for finding the point of intersection.

Solving Linear Equations

Solving linear equations is a basic, yet vital skill in finding intersections and other geometric solutions. Once we've completed the substitution, the task boils down to solving a simple linear equation in \(t\).
This follows standard methods of isolating the variable to find its specific value. In our exercise, after substitution, we end up with:

• Combine and simplify terms: \[13 + 5t = 8\]
• Solve for \(t\) by subtracting 13: \[5t = 8 - 13\]
• Obtain \(t\) by dividing by 5: \[t = -1\]

With \(t = -1\), we can plug this value back into the parametric equations to find the intersection point \((3, 1, 1)\). Solving these linear equations efficiently uncovers the vital point where geometry unfolds.